Numerical approximation of Young measures in non-convex variational problems
Consider the problem \[ {\text{min }}\Phi (u),\qquad \text{s.t. } u\in W^{1,p}(\Omega),\quad |u|_{\partial\Omega } =u_D,\tag{P} \] \noindent where \(\Phi (u):=\int_{\Omega}F(x,\nabla u(x))+G(x,u(x)) dx\). \(W^{1,p}(\Omega)\) denotes the Sobolev space \(\{u: \Omega\rightarrow \mathbb{R}; (u,\nabla u)\in L^p(\Omega; \mathbb{R}^{n+1})\}\), \(u_D\in W^{1-1/p, p}\) determines the prescribed trace of \(u\) on the boundary \(\partial\Omega\) of a bounded Lipschitz domain \(\Omega\subset \mathbb{R}^n\). \(L^p(\Omega; \mathbb{R}^{n})\) is a Lebesgue space of measurable functions \(y:\Omega\rightarrow \mathbb{R}^n\) with \(\|y\|:=(\int_{\Omega}|y(x)|^p dx)^{1/p}<+\infty\), \(1<p<+\infty \). Suppose \(F: \Omega\times \mathbb{R}^n\rightarrow \mathbb{R}\), \(G: \Omega\times\mathbb{R}\rightarrow \mathbb{R}\) are Carathéodory functions satisfying \(c|s|^p-C\leq F(x,s)\leq C(1+|s|^p)\) and \(|G(x,u)|\leq a(x)+C|u|^q\) for some \(a\in L^1(\Omega)\), \(c>0\), \(C\in\mathbb{R}\), \(1<q<{pn/(n-p)}\) (\(q<+\infty\) if \(p\geq n\)). Then \(\Phi :W^{1,p}(\Omega)\rightarrow \mathbb{R}\) is well defined, continuous, coercive, but not weakly lower semi-continuous if \(F(x,.)\) is non-convex. It causes accelerated oscillations of the gradient of minimizing sequence of (P) and a failure of existence of a solution to (P). One way to overcome this drawback, which follows also this paper, is to relax the problem by using the continuous extension of \(\Phi\) on a suitable hull of the original space \(W^{1,p}(\Omega)\). The paper presents a relaxation and approximation theory together with optimality conditions for the problem (P) by using the Young measures (Y.m.) which are usually composed from a few atoms. An active-set type algorithm based on the Y.m. is given. The support of the atoms is estimated from the Weierstrass maximum principle involving a Hamiltonian whose good guess is obtained by a multilevel technique. Numerical experiments on a one-dimensional variational problem are supplied.
scientific article; zbMATH DE number 1419741